1976 LA 360 to 1972 A727 tranny.

[Director]

What type torque converter and flex plate is needed to balance la360 to727 transmission? The 360 is externally balanced and i have not found the proper combination to stop vibrations.

 

[Xcudame]

Make sure you have a 360 harmonic balancer on the front of the crankshaft. The 360 has extra weight on one side of the balancer. Here's one from Summit.
DODGE Summit Racing SUM-163318 Summit Racing™ OE-Style Harmonic Dampers | Summit Racing
Here's a special 360 flex plate so you can use a "neutral" torque converter (no extra weights).
DODGE B&M 10236 B&M Flexplates | Summit Racing

They made weighted torque converters, but it's easier to use the B&M flex plate.

 

[Bret Schneider]

Like Xcudame just said, along with the balancer either the torque converter or the flex plate needs to be weighted. Just make sure it's one or the other and you don't somehow end up with weights on both of them.